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Learning to be Bayesian without supervision
 in Adv. Neural Information Processing Systems (NIPS*06
, 2007
"... Bayesian estimators are defined in terms of the posterior distribution. Typically, this is written as the product of the likelihood function and a prior probability density, both of which are assumed to be known. But in many situations, the prior density is not known, and is difficult to learn from ..."
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Cited by 19 (8 self)
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Bayesian estimators are defined in terms of the posterior distribution. Typically, this is written as the product of the likelihood function and a prior probability density, both of which are assumed to be known. But in many situations, the prior density is not known, and is difficult to learn from data since one does not have access to uncorrupted samples of the variable being estimated. We show that for a wide variety of observation models, the Bayes least squares (BLS) estimator may be formulated without explicit reference to the prior. Specifically, we derive a direct expression for the estimator, and a related expression for the mean squared estimation error, both in terms of the density of the observed measurements. Each of these priorfree formulations allows us to approximate the estimator given a sufficient amount of observed data. We use the first form to develop practical nonparametric approximations of BLS estimators for several different observation processes, and the second form to develop a parametric family of estimators for use in the additive Gaussian noise case. We examine the empirical performance of these estimators as a function of the amount of observed data. 1
Optimal Estimation in Sensory Systems
, 2009
"... Abstract: A variety of experimental studies suggest that sensory systems are capable of performing estimation or decision tasks at nearoptimal levels. In this chapter, I explore the use of optimal estimation in describing sensory computations in the brain. I define what is meant by optimality and p ..."
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Cited by 6 (5 self)
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Abstract: A variety of experimental studies suggest that sensory systems are capable of performing estimation or decision tasks at nearoptimal levels. In this chapter, I explore the use of optimal estimation in describing sensory computations in the brain. I define what is meant by optimality and provide three quite different methods of obtaining an optimal estimator, each based on different assumptions about the nature of the information that is available to constrain the problem. I then discuss how biological systems might go about computing (and learning to compute) optimal estimates. The brain is awash in sensory signals. How does it interpret these signals, so as to extract meaningful and consistent information about the environment? Many tasks require estimation of environmental parameters, and there is substantial evidence that the system is capable of representing and extracting very precise estimates of these parameters. This is particularly impressive when one considers the fact that the brain is built from a large number of lowenergy unreliable components, whose responses are affected by many extraneous factors (e.g., temperature, hydration, blood glucose and oxygen levels). The problem of optimal estimation is well studied in the statistics and engineering communities, where a plethora of tools have been developed for designing, implementing, calibrating and testing such systems. In recent years, many of these tools have been used to provide benchmarks or models for biological perception. Specifically, the development of signal detection theory led to widespread use of statistical decision theory as a framework for assessing performance in perceptual experiments. More recently, optimal estimation theory (in particular, Bayesian estimation) has been used as a framework for describing human performance in perceptual tasks.
Empirical Bayes least squares estimation without an explicit prior.” NYU Courant Inst
, 2007
"... Bayesian estimators are commonly constructed using an explicit prior model. In many applications, one does not have such a model, and it is difficult to learn since one does not have access to uncorrupted measurements of the variable being estimated. In many cases however, including the case of cont ..."
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Cited by 4 (4 self)
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Bayesian estimators are commonly constructed using an explicit prior model. In many applications, one does not have such a model, and it is difficult to learn since one does not have access to uncorrupted measurements of the variable being estimated. In many cases however, including the case of contamination with additive Gaussian noise, the Bayesian least squares estimator can be formulated directly in terms of the distribution of noisy measurements. We demonstrate the use of this formulation in removing noise from photographic images. We use a local approximation of the noisy measurement distribution by exponentials over adaptively chosen intervals, and derive an estimator from this approximate distribution. We demonstrate through simulations that this adaptive Bayesian estimator performs as well or better than previously published estimators based on simple prior models. 1
Optimal estimation: Prior free methods and physiological application
 Ph.D. dissertation, Courant Institute of Mathematical Sciences
, 2007
"... First and foremost, I would like to thank my advisors, Eero Simoncelli and Dan Tranchina. Dan supervised my work on cortical modeling, and his insight and advice were extremely helpful in carrying out the bulk of the work of Chapter 1. He also had many useful comments about the remainder of the mate ..."
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Cited by 2 (2 self)
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First and foremost, I would like to thank my advisors, Eero Simoncelli and Dan Tranchina. Dan supervised my work on cortical modeling, and his insight and advice were extremely helpful in carrying out the bulk of the work of Chapter 1. He also had many useful comments about the remainder of the material in the thesis. Over the years, I have learned a lot about computational neuroscience in general from discussions with him. Eero supervised my work on priorfree methods and applications, which make up the substance of Chapters 24. His intuition, insight and ideas were crucial in helping me progress in this line of research, and more importantly, in obtaining useful results. I also learned a lot from him about image processing, statistics and computational neuroscience, amongst other things. I would like to thank my third reader, Charlie Peskin, for his input to my thesis and defense and helpful discussions about the material. I would also like to thank Mehryar Mohri for being on my committee and for some useful discussions about VC type bounds for regression. As well, I would like to thank Francesca Chiaromonte for being on my committee, and for helpful discussions and comments about the material in the thesis. It was good to have a statistician’s point of view on the work. I would like to thank Bob Shapley for his helpful input, and for information about contrast dependent summation area. I would also like to thank him for letting me sit in on his ”new view ” class about visual cortex, where I read some very useful papers. I would like to thank members of the Laboratory for Computational v Vision, for helpful comments and discussions along the way. I would also like to thank LCV alumni Liam Paninski and Jonathan Pillow, who both had some particularly useful comments about the priorfree methods. I would also like thank the various people at Courant, too numerous to mention, who have provided help along the way.
Learning least squares estimators without assumed priors or supervision
, 2009
"... The two standard methods of obtaining a leastsquares optimal estimator are (1) Bayesian estimation, in which one assumes a prior distribution on the true values and combines this with a model of the measurement process to obtain an optimal estimator, and (2) supervised regression, in which one opti ..."
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Cited by 2 (1 self)
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The two standard methods of obtaining a leastsquares optimal estimator are (1) Bayesian estimation, in which one assumes a prior distribution on the true values and combines this with a model of the measurement process to obtain an optimal estimator, and (2) supervised regression, in which one optimizes a parametric estimator over a training set containing pairs of corrupted measurements and their associated true values. But many realworld systems do not have access to either supervised training examples or a prior model. Here, we study the problem of obtaining an optimal estimator given a measurement process with known statistics, and a set of corrupted measurements of random values drawn from an unknown prior. We develop a general form of nonparametric empirical Bayesian estimator that is written as a direct function of the measurement density, with no explicit reference to the prior. We study the observation conditions under which such “priorfree ” estimators may be obtained, and we derive specific forms for a variety of different corruption processes. Each of these priorfree estimators may also be used to express the mean squared estimation error as an expectation over the measurement density, thus generalizing Stein’s unbiased risk estimator (SURE) which provides such an expression for the additive Gaussian noise case. Minimizing this expression over measurement samples provides an “unsupervised
An Empirical Bayesian interpretation and generalization of NLmeans
"... A number of recent algorithms in signal and image processing are based on the empirical distribution of localized patches. Here, we develop a nonparametric empirical Bayesian estimator for recovering an image corrupted by additive Gaussian noise, based on fitting the density over image patches with ..."
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Cited by 1 (1 self)
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A number of recent algorithms in signal and image processing are based on the empirical distribution of localized patches. Here, we develop a nonparametric empirical Bayesian estimator for recovering an image corrupted by additive Gaussian noise, based on fitting the density over image patches with a local exponential model. The resulting solution is in the form of an adaptively weighted average of the observed patch with the mean of a set of similar patches, and thus both justifies and generalizes the recently proposed nonlocalmeans (NLmeans) method for image denoising. Unlike NLmeans, our estimator includes a dependency on the size of the patch similarity neighborhood, and we show that this neighborhood size can be chosen in such a way that the estimator converges to the optimal Bayes least squares estimator as the amount of data grows. We demonstrate the increase in performance of our method compared to NLmeans on a set of simulated examples. 1
ARTICLE Communicated by Konrad Paul Kording Least Squares Estimation Without Priors or Supervision
"... Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values) or a prior probability model for the true values. Here, we consider the problem of obtaining a least squares estimator given a measurement process with kn ..."
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Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values) or a prior probability model for the true values. Here, we consider the problem of obtaining a least squares estimator given a measurement process with known statistics (i.e., a likelihood function) and a set of unsupervised measurements, each arising from a corresponding true value drawn randomly from an unknown distribution. We develop a general expression for a nonparametric empirical Bayes least squares (NEBLS) estimator, which expresses the optimal least squares estimator in terms of the measurement density, with no explicit reference to the unknown (prior) density. We study the conditions under which such estimators exist and derive specific forms for a variety of different measurement processes. We further show that each of these NEBLS estimators may be used to express the mean squared estimation error as an expectation over the measurement density alone, thus generalizing Stein’s unbiased